Need help determining the bending moment acting on the column. At the top of the column, the cantilever tube is slid into a shorter piece of tubing. The shorter piece of tubing is welded to the vertical tube. I know for eccentrically loaded columns the eccentric load can be resolved to a centric force and moment. Though all of the examples I have seen for columns with an eccentric load, the cantilever part (such as a bracket) has been welded or bolted to the column. Can I the assume the bending moment is simply the load times the distance from the load to the center of the column?
Assuming the force is placed at a distance L from the support, then the moment of the Force P with respect to the foundation will be $ M = P\cdot L$.
If the horizontal square tubing tilts by an angle $ theta$ then the will be equal to $M = O \cdot L\cdot \cos\theta$
For angle up to 10 degrees, the $\cos \theta$ has a minimal change ($\cos 10^o =0.984$), so for most intents and purposes it doesn't change.
The crucial bit is if the horizontal tubing can slide. In that case things will be different.
Q: Can I assume the bending moment is simply the load times the distance from the load to the center of the column?
A: Yes. You have to make the assumption that the cantilever and the short tubing are fitted tightly, so the resultant forces remain in the same locations throughout the process. That is, there are no excessive geometry changes at the beam-tubing interface. (I suggest you consider welding the beam to the tubing).
Q: Also, would the column length for the purpose of calculation include the height of the cantilevered tube?
A: Yes, you can include the height of the short tubing, but you shouldn't, because it complicates the process. Rather, you shall think the loading path - the cantilever causing the forces (P & M) on the short tubing, then the forces are passed to the column (below the tubing) through the welds at the column-tubing interface.
For stress check and design, you have 3 components to consider - the column, the cantilever beam, and the beam-column connection (the short tubing and welds).
For the load is dynamic in nature and frequent use - weld a cap plate to the end of the beam to prevent incidental movement. Then shim at the point "a" if necessary.
For static load application and infrequent cycle of loading-unloading, shim at the point "a", or both points "a" and "b" should be adequate to ascertain the full point contact as assumed and calculated.
Another suggestion for light applications. Note the wall of the cantilever will govern the magnitude of the applied load.
Even if the connection between the cantilever tube and the short tubing is loose and the cantilever tube will rotate a bit before it engages the bracket, it still works and imparts the same moment as long as the rotation doesn't cause it to slide off. You'd be surprised that many lifting jacks count on the biting force of a miss-aligned connection to work.
WE would measure the column height, not including the short tubing. The column is considered a column as long as there is continuity of the section along the length, so the longitudinal fibers can strain and take stress continuously.
If the short tube is not stiff enough to be considered rigid wrt. with other parts, we should try to analyze its deformation and add it to the mechanism.
When you resolve the axial load and moment you can replace the two with an off-center axial load. in many situations, it is recommended to pick a column size with a baseplate where this off-center load falls within 1/3 of the core of the column.